The purpose of this research project was to develop the theory describing the topological structure of the dynamical systems. This was done by extending the Conley index theory, which was originally formulated for gradient-like systems, to a broader class of dynamical systems especially the ones exhibiting chaos, a recurrent and complex behavior in their dynamics. The main results of this project are summarized in the following three items:1. Development of the Conley index theory adapted for singularly perturbed vector fields, and its application to the analysis of some chaotic dynamical systems: In case that the singularly perturbed vector filed has a one-dimensional slow manifold, its phase space structure can be decomposed into the form of the tube-box-cap collection, which enables us to obtain the Conley index information of the entire phase space structure from the analysis of the individual peices of the decomposition. As a result, one can obtain the properties of the characteri
… Morestic orbits, such as periodic and connecting orbits. As an application, the theory was tested by analyzing the model differential equation of a irregular oscillatory behavior of a shallow water wave, and concluded that the behavior is chaotic.2. Extension of the transition matrix theory for multi-parameter systems: The notion of transition matrix is re-considered, which resulted in a new axiomatic formulation, namely, the transition matrix is a chain map on the chain complex obtained from the homology Conley indices given by the Morse decomposition. This new formulation can be used to naturally extend the notion of transition matrix for multi-parameter families of dynamical systems.3. Other related results: For a piecewise linear one dimensional maps, we studied topoplogical entropy which can be a kind of measurement of the complexity of the dynamical systems. Related to this argument, a study of rigorous proof for the existence of chaotic attracter with computer aid is now in progress. Less